Difference between revisions of "FND-STA-Probability distribution"

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<div style="padding:5px; border:1px solid #000000; background-color:#b3dbce; font-size:300%; font-weight:400; color: #000000; width:100%;">
 
Probability Distribution
 
Probability Distribution
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(Nature of a probability distribution, important distributions, comparing observed and simulated probability distributions, Kullback-Leibler diveregence, the Kolmogorov-Smirnov test.)
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<b>Keywords:</b>&nbsp;
 
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Nature of a probability distribution, important distributions, comparing observed and simulated probability distributions, Kullback-Leibler diveregence, the Kolmogorov-Smirnov test.
 
 
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<b>Abstract:</b><br />
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== Abstract ==
 
 
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<section begin=abstract />
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<!-- included from "./components/FND-STA-Probability_distribution.components.txt", section: "abstract" -->
 
 
Probability distributions are at the core of any statistical analysis, in which modelled distributions are compared with sampled distributions to relate an observation to our theoretical understanding. This unit introduces the principles, discusses Poisson, uniform, and normal distributions, and presents methods to compare distributions with each other and quantify the difference.
 
Probability distributions are at the core of any statistical analysis, in which modelled distributions are compared with sampled distributions to relate an observation to our theoretical understanding. This unit introduces the principles, discusses Poisson, uniform, and normal distributions, and presents methods to compare distributions with each other and quantify the difference.
 
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<!-- ============================ -->
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== This unit ... ==
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=== Prerequisites ===
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<!-- included from "./components/FND-STA-Probability_distribution.components.txt", section: "prerequisites" -->
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<b>Objectives:</b><br />
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<!-- included from "./data/ABC-unit_components.txt", section: "notes-external_prerequisites" -->
 
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You need the following preparation before beginning this unit. If you are not familiar with this material from courses you took previously, you need to prepare yourself from other information sources:
 
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<!-- included from "./data/ABC-unit_prerequisites.txt", section: "calculus" -->
 
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*<b>Calculus</b>: functions and equations; polynomial functions, logarithms, trigonometric functions; integrals and derivatives; theorem and proof.
 
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<!-- included from "./data/ABC-unit_components.txt", section: "notes-prerequisites" -->
 
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You need to complete the following units before beginning this one:
 
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*[[FND-STA-Probability|FND-STA-Probability (Probability)]]
 
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=== Objectives ===
 
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<!-- included from "./components/FND-STA-Probability_distribution.components.txt", section: "objectives" -->
 
 
This unit will ...
 
This unit will ...
 
* ... introduce basic concepts of probability distributions;
 
* ... introduce basic concepts of probability distributions;
 
* ... demonstrate the Poisson, the uniform, and the normal distribution;
 
* ... demonstrate the Poisson, the uniform, and the normal distribution;
 
* ... teach how to visually and quantitatively compare them.
 
* ... teach how to visually and quantitatively compare them.
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<b>Outcomes:</b><br />
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=== Outcomes ===
 
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<!-- included from "./components/FND-STA-Probability_distribution.components.txt", section: "outcomes" -->
 
 
After working through this unit you ...
 
After working through this unit you ...
 
* ... can interpret observed events in terms of probability distributions;
 
* ... can interpret observed events in terms of probability distributions;
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* ... can compare observed distributions against each other with <code>qqplot()</code>.
 
* ... can compare observed distributions against each other with <code>qqplot()</code>.
 
* ... can use Kullback-Leibler divergence for discrete distributions, and <code>ks.test()</code> for continuous distributions to quantify differences.
 
* ... can use Kullback-Leibler divergence for discrete distributions, and <code>ks.test()</code> for continuous distributions to quantify differences.
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<!-- ============================ -->
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=== Deliverables ===
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<hr>
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<!-- included from "./components/FND-STA-Probability_distribution.components.txt", section: "deliverables" -->
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<b>Deliverables:</b><br />
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<!-- included from "./data/ABC-unit_components.txt", section: "deliverables-time_management" -->
 
<!-- included from "./data/ABC-unit_components.txt", section: "deliverables-time_management" -->
 
*<b>Time management</b>: Before you begin, estimate how long it will take you to complete this unit. Then, record in your course journal: the number of hours you estimated, the number of hours you worked on the unit, and the amount of time that passed between start and completion of this unit.
 
*<b>Time management</b>: Before you begin, estimate how long it will take you to complete this unit. Then, record in your course journal: the number of hours you estimated, the number of hours you worked on the unit, and the amount of time that passed between start and completion of this unit.
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<!-- included from "./data/ABC-unit_components.txt", section: "deliverables-insights" -->
 
<!-- included from "./data/ABC-unit_components.txt", section: "deliverables-insights" -->
 
*<b>Insights</b>: If you find something particularly noteworthy about this unit, make a note in your [[ABC-Insights|'''insights!''' page]].
 
*<b>Insights</b>: If you find something particularly noteworthy about this unit, make a note in your [[ABC-Insights|'''insights!''' page]].
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<section end=deliverables />
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<!-- ============================  -->
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<hr>
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<section begin=prerequisites />
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<b>Prerequisites:</b><br />
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<!-- included from "./data/ABC-unit_components.txt", section: "notes-external_prerequisites" -->
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You need the following preparation before beginning this unit. If you are not familiar with this material from courses you took previously, you need to prepare yourself from other information sources:
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<!-- included from "./data/ABC-unit_prerequisites.txt", section: "calculus" -->
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*<b>Calculus</b>: functions and equations; polynomial functions, logarithms, trigonometric functions; integrals and derivatives; theorem and proof.
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<!-- included from "./data/ABC-unit_components.txt", section: "notes-prerequisites" -->
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This unit builds on material covered in the following prerequisite units:
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*[[FND-STA-Probability|FND-STA-Probability (Probability)]]
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<section end=prerequisites />
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== Contents ==
 
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{{ABC-unit|FND-STA-Probability_distribution.R}}
 
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== Further reading, links and resources ==
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== Contents ==
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<!-- {{#pmid: 19957275}} -->
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<!-- included from "./components/FND-STA-Probability_distribution.components.txt", section: "contents" -->
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<!-- {{WWW|WWW_GMOD}} -->
 
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<!-- <div class="reference-box">[http://www.ncbi.nlm.nih.gov]</div> -->
 
  
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== Notes ==
 
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<references />
 
  
 
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== Self-evaluation ==
 
== Self-evaluation ==
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<!-- included from "./data/ABC-unit_components.txt", section: "notes" -->
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<references />
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== Further reading, links and resources ==
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<!-- {{#pmid: 19957275}} -->
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<!-- {{WWW|WWW_GMOD}} -->
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<!-- <div class="reference-box">[http://www.ncbi.nlm.nih.gov]</div> -->
  
 
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Revision as of 19:32, 26 January 2018

Probability Distribution

(Nature of a probability distribution, important distributions, comparing observed and simulated probability distributions, Kullback-Leibler diveregence, the Kolmogorov-Smirnov test.)


 


Abstract:

Probability distributions are at the core of any statistical analysis, in which modelled distributions are compared with sampled distributions to relate an observation to our theoretical understanding. This unit introduces the principles, discusses Poisson, uniform, and normal distributions, and presents methods to compare distributions with each other and quantify the difference.

Objectives:
This unit will ...

  • ... introduce basic concepts of probability distributions;
  • ... demonstrate the Poisson, the uniform, and the normal distribution;
  • ... teach how to visually and quantitatively compare them.

Outcomes:
After working through this unit you ...

  • ... can interpret observed events in terms of probability distributions;
  • ... are familar with the Poisson, the uniform, and the normal distribution;
  • ... can compare observed distributions against the normal distribution with qqnorm().
  • ... can compare observed distributions against each other with qqplot().
  • ... can use Kullback-Leibler divergence for discrete distributions, and ks.test() for continuous distributions to quantify differences.

Deliverables:

  • Time management: Before you begin, estimate how long it will take you to complete this unit. Then, record in your course journal: the number of hours you estimated, the number of hours you worked on the unit, and the amount of time that passed between start and completion of this unit.
  • Journal: Document your progress in your Course Journal. Some tasks may ask you to include specific items in your journal. Don't overlook these.
  • Insights: If you find something particularly noteworthy about this unit, make a note in your insights! page.

Prerequisites:
You need the following preparation before beginning this unit. If you are not familiar with this material from courses you took previously, you need to prepare yourself from other information sources:

  • Calculus: functions and equations; polynomial functions, logarithms, trigonometric functions; integrals and derivatives; theorem and proof.

This unit builds on material covered in the following prerequisite units:


 



 



 


Contents

 

Task:

 
  • Open RStudio and load the ABC-units R project. If you have loaded it before, choose File → Recent projects → ABC-Units. If you have not loaded it before, follow the instructions in the RPR-Introduction unit.
  • Choose Tools → Version Control → Pull Branches to fetch the most recent version of the project from its GitHub repository with all changes and bug fixes included.
  • Type init() if requested.
  • Open the file FND-STA-Probability_distribution.R and follow the instructions.


 

Note: take care that you understand all of the code in the script. Evaluation in this course is cumulative and you may be asked to explain any part of code.


 


 

Self-evaluation

Notes

Further reading, links and resources

 



 

If in doubt, ask! If anything about this learning unit is not clear to you, do not proceed blindly but ask for clarification. Post your question on the course mailing list: others are likely to have similar problems. Or send an email to your instructor.


 

About ...
 
Author:

Boris Steipe <boris.steipe@utoronto.ca>

Created:

2017-08-05

Modified:

2017-08-05

Version:

1.0

Version history:

  • 1.0 New material

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